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Mixed Number and Fraction Estimation

Estimating sums and differences

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Mixed Number and Fraction Estimation

Let’s Think About It

Credit: Theresa Thompson
Source: https://www.flickr.com/photos/theresasthompson/2999130055/
License: CC BY-NC 3.0

Ron is running for student council president. He has been campaigning for three weeks. Yesterday he and 2 of his friends handed out stickers to every student in his grade who was at school. Later in the day he noticed that 58 of the 130 students in his grade were wearing one of his stickers. How can Ron figure out approximately what fraction of the students in his grade are wearing one of his stickers?

In this concept, you will learn to approximate fractions and mixed numbers using common benchmarks.

Guidance

Recall that a fraction describes a part of a whole number. The number written below the bar in a fraction is the denominator. The denominator tells you how many parts the whole is divided into. The numerator is the number above the bar in a fraction. The numerator tells you how many parts of the whole you have.

Because a whole can be divided into any number of parts, it is sometimes difficult to get a good sense of the value of a fraction when the denominator of the fraction is large. In order to get an approximate idea of the value of a fraction, you can compare the fraction with three benchmarks: 0, \begin{align*}\frac{1}{2}\end{align*}, and 1. To use the fraction benchmarks, try to determine which benchmark the given fraction is closest to.

Here is an example.

What is the approximate size of \begin{align*}\frac{17}{18}\end{align*}?

First, use the benchmarks. Determine whether the fraction is closest to 0, \begin{align*}\frac{1}{2}\end{align*}, or 1.

Notice that the denominator is 18 and the numerator is 17. This means you have 17 out of 18 parts. 18 out of 18 parts would be 1 whole, so 17 out of 18 is close to 1.

The answer is \begin{align*}\frac{17}{18}\end{align*} is approximately 1.

Keep in mind that \begin{align*}\frac{17}{18}\end{align*} is only approximately equal to 1. When you are trying to imagine how much \begin{align*}\frac{17}{18}\end{align*} is, it can be helpful to think of it as being about equal to 1.

Here is another example.

What is the common benchmark for \begin{align*}\frac{24}{49}\end{align*}?

First, look at the relationship between the numerator and the denominator. The numerator of 24 is a little less than half of the denominator of 49. This means \begin{align*}\frac{24}{49}\end{align*} is close to \begin{align*}\frac{1}{2}\end{align*}.

The answer is \begin{align*}\frac{1}{2}\end{align*} is an appropriate benchmark for \begin{align*}\frac{24}{49}\end{align*}.

You can also use fraction benchmarks when working with mixed numbers. Instead of 0, \begin{align*}\frac{1}{2}\end{align*}, and 1, the benchmarks will be based on the whole number part of the mixed number.

Here is an example.

What is the benchmark for \begin{align*}7 \frac{1}{8}\end{align*}?

First, note that this is a mixed number. The benchmarks to consider are 7, \begin{align*}7 \frac{1}{2}\end{align*}, and 8.

Next, look at the fraction part of the mixed number, \begin{align*}\frac{1}{8}\end{align*}. Compare the numerator with the denominator. The numerator of 1 is much less than the denominator of 8. You only have 1 part out of 8. This means \begin{align*}\frac{1}{8}\end{align*} is close to 0. So \begin{align*}7 \frac{1}{8}\end{align*} is close to 7.

The answer is 7 is an appropriate benchmark for \begin{align*}7 \frac{1}{8}\end{align*}.

Guided Practice

Name the common benchmark for the fraction \begin{align*}\frac{4}{7}\end{align*}.

First, compare the numerator with the denominator. The numerator is 4 and the denominator is 7. 4 is a little more than half of 7. This means \begin{align*}\frac{4}{7}\end{align*} is close to \begin{align*}\frac{1}{2}\end{align*}.

The answer is \begin{align*}\frac{1}{2}\end{align*} is an appropriate benchmark for \begin{align*}\frac{4}{7}\end{align*}.

Examples

Example 1

Name the common benchmark for the fraction \begin{align*}\frac{1}{12}\end{align*}.

First, look at the relationship between the numerator and the denominator. The numerator of 1 is much less than the denominator of 12. You only have 1 part out of 12. This means \begin{align*}\frac{1}{12}\end{align*} is close to 0.

The answer is 0 is an appropriate benchmark for \begin{align*}\frac{1}{12}\end{align*}.

Example 2

Name the common benchmark for the fraction \begin{align*}\frac{5}{6}\end{align*}.

First, look at the relationship between the numerator and the denominator. Notice that the denominator is 6 and the numerator is 5. This means you have 5 out of 6 parts. 6 out of 6 parts would be 1 whole, so 5 out of 6 is close to 1.

The answer is \begin{align*}\frac{5}{6}\end{align*} is approximately 1.

Example 3

Name the common benchmark for the mixed number \begin{align*}9 \frac{3}{9}\end{align*}.

First, note that this is a mixed number. The benchmarks to consider are 9, \begin{align*}9 \frac{1}{2}\end{align*}, and 10.

Next, look at the fraction part of the mixed number, \begin{align*}\frac{3}{9}\end{align*}. Compare the numerator with the denominator. The numerator of 3 is a little less than half the denominator of 9. This means \begin{align*}\frac{3}{9}\end{align*} is close to \begin{align*}\frac{1}{2}\end{align*}. So \begin{align*}9 \frac{3}{9}\end{align*} is close to \begin{align*}9 \frac{1}{2}\end{align*}.

The answer is \begin{align*}9 \frac{1}{2}\end{align*} is an appropriate benchmark for \begin{align*}9 \frac{3}{9}\end{align*}.

Follow Up

Credit: Alan Cleaver
Source: https://www.flickr.com/photos/alancleaver/4446461866/
License: CC BY-NC 3.0

Remember Ron who is running for student council president? \begin{align*}\frac{58}{130}\end{align*} of the students in his grade are wearing the stickers he handed out. Ron wants to know approximately what fraction of the students in his grade are wearing a sticker.

First, Ron should compare the numerator with the denominator. The numerator is 58 and the denominator is 130. 58 is a little less than half of 130. This means \begin{align*}\frac{58}{130}\end{align*} is close to \begin{align*}\frac{1}{2}\end{align*}.

The answer is approximately \begin{align*}\frac{1}{2}\end{align*} the students in Ron’s grade are wearing a sticker.

Explore More

Approximate the value of the following fractions using the benchmarks 0, \begin{align*}\frac{1}{2}\end{align*}, and 1.

1. \begin{align*}\frac{9}{10}\end{align*}

2. \begin{align*}\frac{11}{20}\end{align*}

3. \begin{align*}\frac{2}{32}\end{align*}

4. \begin{align*}\frac{21}{22}\end{align*}

5. \begin{align*}\frac{1}{23}\end{align*}

6. \begin{align*}\frac{11}{100}\end{align*}

7. \begin{align*}\frac{2}{3}\end{align*}

8. \begin{align*}\frac{14}{28}\end{align*}

9. \begin{align*}\frac{16}{30}\end{align*}

10. \begin{align*}\frac{18}{21}\end{align*}

Approximate the value of the following mixed numbers.

11. \begin{align*}2 \frac{79}{80}\end{align*}

12. \begin{align*}6 \frac{1}{10}\end{align*}

13. \begin{align*}43 \frac{7}{15}\end{align*}

14. \begin{align*}8 \frac{7}{99}\end{align*}

15. \begin{align*}6 \frac{21}{22}\end{align*}

Vocabulary

Denominator

Denominator

The denominator of a fraction (rational number) is the number on the bottom and indicates the total number of equal parts in the whole or the group. \frac{5}{8} has denominator 8.
Equivalent

Equivalent

Equivalent means equal in value or meaning.
Equivalent Fractions

Equivalent Fractions

Equivalent fractions are fractions that can each be simplified to the same fraction. An equivalent fraction is created by multiplying both the numerator and denominator of the original fraction by the same number.
fraction

fraction

A fraction is a part of a whole. A fraction is written mathematically as one value on top of another, separated by a fraction bar. It is also called a rational number.
Greatest Common Factor

Greatest Common Factor

The greatest common factor of two numbers is the greatest number that both of the original numbers can be divided by evenly.
improper fraction

improper fraction

An improper fraction is a fraction in which the absolute value of the numerator is greater than the absolute value of the denominator.
Mixed Number

Mixed Number

A mixed number is a number made up of a whole number and a fraction, such as 4\frac{3}{5}.
Numerator

Numerator

The numerator is the number above the fraction bar in a fraction.
Simplify

Simplify

To simplify means to rewrite an expression to make it as "simple" as possible. You can simplify by removing parentheses, combining like terms, or reducing fractions.
Whole Numbers

Whole Numbers

The whole numbers are all positive counting numbers and zero. The whole numbers are 0, 1, 2, 3, ...

Image Attributions

  1. [1]^ Credit: Theresa Thompson; Source: https://www.flickr.com/photos/theresasthompson/2999130055/; License: CC BY-NC 3.0
  2. [2]^ Credit: Alan Cleaver; Source: https://www.flickr.com/photos/alancleaver/4446461866/; License: CC BY-NC 3.0

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